Preconditioned Conjugate Gradient Technique Research Articles

GPU-based parallel computation for discontinuous deformation analysis (DDA) method and its application to modelling earthquake-induced landslide

Graphic Processing Unit (GPU), as a computing device, has upgraded from single-subject graphical processors to multi-core processors with tremendous computational horsepower. This paper proposes to accelerate the DDA using parallel Jacobi Preconditioned Conjugate Gradient (JPCG) technique on GPUs. Based on the results of two numerical examples, the calculation accuracies of the DDA with serial and parallel solvers are validated, and we found that the DDA with parallel solvers exhibits a much higher execution efficiency. The movement process of Daguangbao landslide triggered by the Wenchuan earthquake is replicated and the modeled deposit pattern coincides well with the actual topography after earthquake.

Numerical approximation of solitary waves of the Benjamin equation

This paper presents several numerical techniques to generate solitary-wave profiles of the Benjamin equation. The formulation and implementation of the methods focus on some specific points of the problem: on the one hand, the approximation of the nonlocal term is accomplished by Fourier techniques, which determine the spatial discretization used in the experiments. On the other hand, in the numerical continuation procedure suggested by the derivation of the model and already discussed in the literature, several algorithms for solving the nonlinear systems are described and implemented: the Petviashvili method, the Preconditioned Conjugate Gradient Newton method and two Squared-Operator methods. A comparative study of these algorithms is made in the case of the Benjamin equation; Newton's method combined with Preconditioned Conjugate Gradient techniques, emerges as the most efficient. The resulting numerical profiles are shown to have a high accuracy as travelling-wave solutions when they are used as initial conditions in a time-stepping procedure for the Benjamin equation. The paper also explores the generation of multi-pulse solitary waves.

Deflated preconditioned conjugate gradient solvers for the pressure‐Poisson equation: Extensions and improvements

AbstractExtensions and improvements to a deflated preconditioned conjugate gradient technique for the solution of the pressure‐Poisson equation within an incompressible flow solver are described. In particular, the use of the technique for embedded grids, for cases where volume of fluid or level set schemes are required and its implementation on parallel machines are considered. Several examples are included that demonstrate a considerable reduction in the number of iterations and a remarkable insensitivity to the number of groups/ regions chosen and/or to the way the groups are formed. Copyright © 2010 John Wiley & Sons, Ltd.

Deflated preconditioned conjugate gradient solvers for the Pressure–Poisson equation

A deflated preconditioned conjugate gradient technique has been developed for the solution of the Pressure–Poisson equation within an incompressible flow solver. The deflation is done using a region-based decomposition of the unknowns, making it extremely simple to implement. The procedure has shown a considerable reduction in the number of iterations. For grids with large graph-depth the savings exceed an order of magnitude. Furthermore, the technique has shown a remarkable insensitivity to the number of groups/regions chosen, and to the way the groups are formed.

Thermal Transient Response of GaAs FETs Under Intentional Electromagnetic Interference (IEMI)

Transient thermal responses of GaAs field-effect transistors (FETs) in the presence of an electromagnetic pulse (EMP) are investigated in this paper. The numerical methodology employed is an efficient nonlinear finite-element method (FEM) that combines the element-by-element FEM and the preconditioned conjugate gradient technique. Parametric studies are carried out to show different pulse parameters on the transient thermal responses as well as maximum channel temperatures of some typical GaAs FETs, with silicon FETs also taken into account for comparison. It is numerically proven that the thermal impact caused by medium EMP will be the most serious compared with fast EMP or ultra-wideband pulse, and the captured maximum channel temperature is proportional to the input power density, approximately of the EMP injected. This research can serve as a base for taking further protection measures to prevent on-chip device from breakdown by the attack of an EMP.

A preconditioned conjugate gradient approach to structural reanalysis for general layout modifications

AbstractThis paper presents a preconditioned conjugate gradient approach to structural static reanalysis for general layout modifications. It is suitable for all types of layout modifications, including the general case in which some original members and nodes are deleted and other new members and nodes are added concurrently. The approach is based on the preconditioned conjugate gradient technique. The preconditioner is constructed, and an efficient implementation for applying the preconditioner is presented, which requires the factorization of the stiffness matrix corresponding to the newly added degrees of freedom only. In particular, the approach can adaptively monitor the accuracy of approximate solutions. Numerical examples show that the condition number of the preconditioned matrix is remarkably reduced. Therefore, the fast convergence and accurate results can be achieved by the approach. Copyright © 2006 John Wiley & Sons, Ltd.

Block adaptive shifting algorithm for filtering based on the Toeplitz preconditioned conjugate gradient technique

We present a new block adaptive algorithm as a variant of the Toeplitz-preconditioned block conjugate gradient (TBCG) algorithm. The proposed algorithm is formulated by combining TBCG algorithm with a data-reusing scheme that is realized by processing blocks of data in an overlapping manner, as in the optimum block adaptive shifting (OBAS) algorithm. Simulation results show that the proposed algorithm is superior to the block conjugate gradient shifting (BCGS), TBCG and Toeplitz-OBAS (TOBAS) algorithms in both convergence rate and tracking property of input signal conditioning.

MPI-based parallel finite element approaches for implicit nonlinear dynamic analysis employing sparse PCG solvers

This paper presents three formulations combining domain decomposition based finite element method with linear preconditioned conjugate gradient (LPCG) technique for solving large-scale problems in structural mechanics on parallel processing machines. In the first formulation called the Global Interface Formulation (GIF), the PCG algorithm is applied on the assembled interface stiffness coefficient matrices of all submeshes. The second formulation called Local Submesh Formulation (LSF) operates on the local unassembled submesh matrices and the preconditioner is constructed using the local submesh information. In the third formulation called Local Interface Formulation (LIF), the sparse PCG algorithm is formulated using the unassembled local schur complement matrices of submeshes. Both diagonal and incomplete Cholesky preconditioners have been employed. These domain decomposition based PCG algorithms have been implemented within a finite element code for nonlinear implicit transient dynamic analysis. Time integration is performed using Newmark-β constant average acceleration method. The parallel finite element code uses an MPI-based message passing approach to provide portable parallel execution on shared, distributed and distributed shared memory computers. Numerical experiments have been conducted on PARAM-10000, an Indian parallel supercomputer to evaluate the performance of the implicit parallel nonlinear finite element code employing the three proposed PCG formulations. Numerical studies indicate that the proposed parallel PCG formulations are highly adaptive for parallel computing and superior in performance when compared to the conventional domain decomposition algorithm with parallel direct solver. The LSF formulation, which is amenable for efficient implementation of communications by way of overlapping with computations found to be superior in performance compared to other two PCG formulations.

Global, exact cosmic microwave background data analysis using Gibbs sampling

We describe an efficient and exact method that enables global Bayesian analysis of cosmic microwave background (CMB) data. The method reveals the joint posterior density (or likelihood for flat priors) of the power spectrum ${C}_{\ensuremath{\ell}}$ and the CMB signal. Foregrounds and instrumental parameters can be simultaneously inferred from the data. The method allows the specification of a wide range of foreground priors. We explicitly show how to propagate the non-Gaussian dependency structure of the ${C}_{\ensuremath{\ell}}$ posterior through to the posterior density of the parameters. If desired, the analysis can be coupled to theoretical (cosmological) priors and can yield the posterior density of cosmological parameter estimates directly from the time-ordered data. The method does not hinge on special assumptions about the survey geometry or noise properties, etc., It is based on a Monte Carlo approach and hence parallelizes trivially. No trace or determinant evaluations are necessary. The feasibility of this approach rests on the ability to solve the systems of linear equations which arise. These are of the same size and computational complexity as the map-making equations. We describe a preconditioned conjugate gradient technique that solves this problem and demonstrate in a numerical example that the computational time required for each Monte Carlo sample scales as ${n}_{p}^{3/2}$ with the number of pixels ${n}_{p}$. We use our method to analyze the data from the Differential Microwave Radiometer on the Cosmic Background Explorer and explore the non-Gaussian joint posterior density of the ${C}_{\ensuremath{\ell}}$ from the Differential Microwave Radiometer on the Cosmic Background Explorer in several projections.

On a one-dimensional optimization problem derived from the efficiency analysis of Newton-PCG-like algorithms

The Newton-PCG (preconditioned conjugate gradient) like algorithms are usually very efficient. However, their efficiency is mainly supported by the numerical experiments. Recently, a new kind of Newton-PCG-like algorithms is derived in (J. Optim. Theory Appl. 105 (2000) 97; Superiority analysis on truncated Newton method with preconditioned conjugate gradient technique for optimization, in preparation) by the efficiency analysis. It is proved from the theoretical point of view that their efficiency is superior to that of Newton's method for the special cases where Newton's method converges with precise Q -order 2 and α (⩾2), respectively. In the process of extending such kind of algorithms to the more general case where Newton's method has no fixed convergence order, the first is to get the solutions to the one-dimensional optimization problems with many different parameter values of α . If these problems were solved by numerical method one by one, the computation cost would reduce the efficiency of the Newton-PCG algorithm, and therefore is unacceptable. In this paper, we overcome the difficulty by deriving an analytic expression of the solution to the one-dimensional optimization problem with respect to the parameter α .

TRANSPORE : A GENERIC HEAT AND MASS TRANSFER COMPUTATIONAL MODEL FOR UNDERSTANDING AND VISUALISING THE DRYING OF POROUS MEDIA

ABSTRACT In this work a sophisticated numerical model is presented that describes the drying of porous media. This model, which is known as TransPore, has evolved over the years through the direct inputs of both authors. Nowadays, TransPore can be used to analyse the drying of media that are of completely arbitrary shape and size, under a variety of drying conditions. The engine of the computational model uses a number of state-of-the-art numerical methods that ensure the simulation results describe the particular drying process accurately, whilst guaranteeing the most efficient and effective usage of computer resources. For example, the numerical discretisation method is based on a completely conservative hybrid finite element control volume technique that uses a finite element mesh for its background gradient interpolation. Furthermore, flux limiting is used to reduce numerical dispersion in the drying kinetics and the generated non-linear system is resolved using the full Newton method for the outer iteration coupled together with a preconditioned conjugate gradient technique for the inner iteration. A graphical interface has been linked to the model to enable online visualisation of the drying process. The mathematical model allows both homogeneous and heterogeneous porous media to be simulated. The resultant software is an extremely powerful and effective tool for investigating existing dryer designs and for proposing new and innovative drying schedules that provide optimal drying quality in minimal drying time.

AUTOMATIC ADAPTIVE 3-D FINITE ELEMENT REFINEMENT USING DIFFERENT‐ORDER TETRAHEDRAL ELEMENTS

Automatic refinement finite element analyses were carried out employing three different-order tetrahedral solid elements for the solution of 3-D stress analysis problems. Numerical results indicated that the adaptive refinement procedure could eliminate effectively the effect of singularities and the optimal convergence rate was achieved in all the examples tested. The preconditioned conjugate gradient technique was used for the solution of the large system of simultaneous equations. By interpolating the initial guess of the iteration solver from the previous converged solution, the number of iterations needed for the solution is lower than expected. Furthermore, when the mesh density distribution pattern has converged, it became even more efficient and independent of the number of degrees of freedom in the finite element mesh. The relative efficiency of the three different-order tetrahedral elements has also been compared in terms of storage and computational cost needed for achieving a certain accuracy. It is found that although the cubic T20 element can achieve the highest convergence rate, the T10 element is the most competitive and effective element in terms of storage and computational cost needed. © 1997 John Wiley & Sons, Ltd.

Automatic adaptive refinement finite element procedure for 3D stress analysis

An automatic adaptive refinement procedure for 3D finite element analysis is presented. Based on a carefully designed procedure to eliminate the instability of the superconvergent patch recovery technique, accurate smoothed stress field for 3D elasticity problem can be established. Well graded adaptive tetrahedral meshes with node spacing compatible withthe specifications as required by the refinement strategy are generated by newly developed mesh refinement scheme. Iterative solution solver employing the preconditioned conjugate gradient technique is used for the solution of large system of simultaneous equations. Practical examples solved by using the quadratic T10 tetrahedral element indicate that the procedure is reliable and effective. Even in the presence of singularities, the optimal convergence rate is achieved and the asymptotic convergence of the error estimator is also observed. Numerical results in this study demonstrate that by combining theoretical derivations with appropriate computation algorithms, it is completely feasible to carry out automatic adaptive analyses even for full 3D problems. Many advantages of the adaptive refinement over the traditional one-pass finite element analysis or uniform refinement are revealed, especially when the exact solution of the problem is difficult and singular points are present inside the problem domain.

Parameterization and efficient implementation of a solvent model for linear-scaling semiempirical quantum mechanical calculations of biological macromolecules

A method is developed to include solvation effects in linear-scaling semiempirical quantum calculations. Favorable scaling of computational effort for large molecules is achieved using a preconditioned conjugate gradient technique in conjunction with a linear-scaling recursive bisection method for evaluation of electrostatic interactions. The method requires approximately 30% computational overhead relative to gas-phase calculations. Effective atomic radii for biological macromolecules are derived from fitting to experimental and theoretical solvation energies for small molecules homologous to amino- and nucleic acid residues.

Ab initio calculations for SiC - Al interfaces: tests of electronic-minimization techniques

Computational techniques for obtaining the electronic ground state in the ab initio pseudopotential method using a plane-wave basis set have been examined for SiC(110) - Al interfaces as metal - ceramic interfaces. The preconditioned conjugate-gradient (CG) method by Teter, Payne and Allan (TPA) performing direct energy minimization is efficient for an SiC surface as it is a non-metallic large system. However, this method encounters oscillations by charge sloshing in the SiC - Al interface because of its metallic nature. The subspace rotation by Gillan instead of the subspace diagonalization improves the convergence behaviour, although it is not easy to remove small oscillations in the present system. We have found that the CG method by Bylander, Kleinman and Lee is more efficient than the other methods for the SiC - Al interface, where the preconditioned CG techniques are used for minimization of energy expectation values in fixed charge density and potential combined with the mixing scheme of Kerker to prevent the charge sloshing. This method is also more efficient than a similar kind of method such as the block Davidson method for the present system, because the band-by-band scheme is more suitable for metal - ceramic interfaces. From the relaxed configurations of the SiC - Al interfaces, we have found strong attractive interactions between C and Al atoms, which should play a favourable role in the adhesion between SiC and Al.

Phase unwrapping by means of multigrid techniques for interferometric SAR

Weighted least squares phase unwrapping is a robust approach to phase unwrapping that unwraps around (rather than through) regions of corrupted phase. Currently, the only practical method for solving the weighted least squares equations is a preconditioned conjugate gradient (PCG) technique. In this paper the authors present a new method for weighted least squares phase unwrapping. Their method is a multigrid technique that solves the equations on smaller, coarser grids by means of Gauss-Seidel relaxation schemes and transfers the intermediate results to the finer grids. A key idea of their approach is to maintain the partial derivatives of the given phase data in separate arrays and to correct these derivatives at the boundaries of the coarser grids. This correction maintains the boundary conditions necessary for convergence to the correct solution. Another key idea of their approach is to transfer the weighting values to the coarser grids in a carefully defined manner. They also present methods for defining the initial phase weights in an automated fashion. The resulting multigrid algorithm converges in only one or two multigrid cycles and is generally 15-25 times faster than the PCG technique.

New PCG based algorithms for the solution of Hermitian Toeplitz systems

In order to solve Toeplitz linear systems An(f)x=b generated by a nonnegative integrable function f, through use of the preconditioned conjugate gradient (PCG) method, several authors have proposed An(g) as preconditioner in the case where g is a trigonometric polynomial [10, 14, 27, 12, 28]. In preceding works, we studied the distribution and the extremal properties of the spectrum of the preconditioned matrix G=An−1(g) An(f). In this paper we prove that the union of the spectra of all the Gn is dense on the essential range of f/g, i.e.,ER(f/g) and we obtain asymptotic information about the rate of convergence of the smallest eigenvalue λln of Gn to r (and of λnn to R). As a consequence of this second order result, it is possible to handle the case where f has zeros of any order θ, through the PCG methods proposed in [10, 14]. This is a noteworthy extension since the techniques developed in [10, 14, 27, 12, 28] are shown to be effective only when f has zeros of even orders. The cost of this procedure is O(n1+c(θ) log n) arithmetic operations (ops) where the quantity c(θ) belongs to interval [0,2−1] and takes the maximum value 2−1 when f has a zero of odd order. Finally, for the special case of zeros of odd orders, we propose a further algorithm which makes use of the PCG techniques proposed in [10, 14, 27, 12, 28] for theeven order case, reducing the cost to O(n long n) ops.

Weighted graph based ordering techniques for preconditioned conjugate gradient methods

We describe the basis of a matrix ordering heuristic for improving the incomplete factorization used in preconditioned conjugate gradient techniques applied to anisotropic PDE's. Several new matrix ordering techniques, derived from well-known algorithms in combinatorial graph theory, which attempt to implement this heuristic, are described. These ordering techniques are tested against a number of matrices arising from linear anisotropic PDE's, and compared with other matrix ordering techniques. A variation of RCM is shown to generally improve the quality of incomplete factorization preconditioners.

Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium.

We present ab initio quantum-mechanical molecular-dynamics simulations of the liquid-metal--amorphous-semiconductor transition in Ge. Our simulations are based on (a) finite-temperature density-functional theory of the one-electron states, (b) exact energy minimization and hence calculation of the exact Hellmann-Feynman forces after each molecular-dynamics step using preconditioned conjugate-gradient techniques, (c) accurate nonlocal pseudopotentials, and (d) Nos\'e dynamics for generating a canonical ensemble. This method gives perfect control of the adiabaticity of the electron-ion ensemble and allows us to perform simulations over more than 30 ps. The computer-generated ensemble describes the structural, dynamic, and electronic properties of liquid and amorphous Ge in very good agreement with experiment. The simulation allows us to study in detail the changes in the structure-property relationship through the metal-semiconductor transition. We report a detailed analysis of the local structural properties and their changes induced by an annealing process. The geometrical, bonding, and spectral properties of defects in the disordered tetrahedral network are investigated and compared with experiment.

On the oblique reflexion and transmission of ocean waves at shore fast sea ice

A mathematical model is reported describing the oblique reflexion and penetration of ocean waves into shore fast sea ice. The arbitrary depth model allows all velocity potentials occurring in the open water region to be matched precisely to their counterparts in the ice-covered region. Matching is done using a preconditioned conjugate gradient technique which allows the complete solution to be found to a predefined precision. The model enables the reflexion and transmission coefficients at the ice edge to be found, and examples are reported for ice plates of different thicknesses. A critical angle is predicted beyond which no travelling wave penetrates the ice sheet; in this case the deflexion of the ice is due only to evanescent modes. Critical angle curves are provided for various ice thicknesses on deep, intermediate and shallow water. The strain field which is set up within the ice sheet due to the incoming waves is also discussed; principal strains are provided as are the strains normal to the ice edge. Finally the spreading function within the ice cover, and some consequences of this function to unimodal seas with realistic open water spreading functions, are reported with the aim of generalizing the work to model the effect of shore fast ice on an incoming directional wave spectrum of specified structure.